Stability of desynchronized 0L-systems (Q794173)

A 0L scheme (\(\Sigma\),g) is defined to be \(\Delta\)-desynchronization of \(S=(\Sigma,h)\) if \(g(a)=\{a,h(a)\}\) for \(a\in \Delta\), \(g(a)=h(a)\) otherwise. For a 0L system \(G=(\Sigma,h,w)\) whose underlying scheme is equal to \(S=(\Sigma,h)\) the \(\Delta\)-desynchronization of S induces \(\Delta\)-desynchronization of G. A system G is defined to be stable under \(\Delta\)-desynchronization w.r.t. a target \(T\subseteq \Sigma^*\) if \(T\cap L(G)=T\cap L(G^ d(\Delta)),\) where L(G) is a language generated by G and \(G^ d(\Delta)\) is \(\Delta\)-desynchronization of G. It is proved that a 0L system is stable under \(\Sigma\)-desynchronization w.r.t. the set of recurrent strings and it is stable under \(\Delta\)- desynchronization w.r.t. the set of adult strings.